Josh Hutcherson - The Journey Series (the third is in development) & The Hunger Games

Michael Fassbender - X-Men & The Alien Series (if you count Prometheus and it's planned two sequels as part of that series)

And then Keria Knightly (Pirates of the Caribbean) had a supporting role in The Phantom Menace, Bill Nighy (Pirates of the Caribbean) appeared in The Deathly Hallows, Sir Michael Cane (The Dark Knight Trilogy) in Jaws 4 & Journey 2, Christian Bale (The Dark Knight Trilogy) in Terminator Salvation, Brendan Fraser (The Mummy) in Journey to the Center of the Earth, Vanessa Hudgens (High School Musical) in Journey 2, Ashley Tisdale (High School Musical) in Scary Movie 5, Robert Pattinson (Twilight) in Goblet of Fire, Sam Claffin (the future Hunger Games) in On Strangers Tide, and then if we're really pushing it Dame Judi Dench (James Bond) cameos in Pirates 4. John Hurt also appears in the Alien, Harry Potter, Hellboy and Indiana Jones series'.

Joe Pesci - Home Alone & Lethal Weapon
Alan Rickman - Die Hard & Harry Potter (Only appeared in one Die Hard, but nobody specified they have to appear in every film )

If you go by the OPs own examples then the actors only need to appear in one of the franchises films to be included - as Jennifer Lawrence was only in the X-men First Class film not the rest.

Chris Evans was in the Fantastic Four franchise as well as the Avenger/Captain America franchise if that is valid?

Christian Bale was in Batman franchise and the Terminator franchise.

Anton Yelchin was also in the Terminator franchise as well as Star Trek franchise.

Matt Damon was in the Bourne franchise and the Ocean's Eleven franchise.

Hugo Weaving is in the Matrix trilogy and the Lord of the Rings and The Transformers franchise not to mention Happy Feet and if we are allowed to include two films as a franchise (not sure we are?) then Babe would also be included

If two film series are included as franchises then I have more - Drew Barrymore for example.

And then again George Clooney (Oceans) & Arnold Schwarzenegger (Terminator) in Batman & Robin; and Rachel Weisz (The Mummy), Jeremy Renner (Marvel), Edward Norton (Marvel) in The Bourne Legacy, with Renner also being in Mission Impossible 4.